Revision notes
A logarithm answers one question — 'what power?' Learn to switch between index form and log form and use the laws of logs.
Tunde knows 2 multiplied by itself some number of times gives 64, but how many times? Indices go forward (26 = 64). Logarithms go backwards: they find the power. That is all a log is!
Since 26 = 64, we say the logarithm of 64 to base 2 is 6, written log2 64 = 6.
So logb N asks: b to what power gives N? For log10 1000, ask '10 to what power is 1,000?' The answer is 3.
The two forms say the same thing: bˣ = N (index form) is the same as logb N = x (log form). The base stays the base; the index becomes the answer.
🔑 Key idea: logb N = x means exactly the same as bˣ = N.
logb b = 1, because b1 = b. For example, log5 5 = 1.
logb 1 = 0, because b0 = 1. For example, log10 1 = 0.
Numbers between 0 and 1 have negative logs: log2 18 = −3, because 2−3 = 18.
🔑 Key idea: logb b = 1 and logb 1 = 0.
Logs are indices, so they follow the index laws. Multiplying numbers means adding their logs: log (a × b) = log a + log b.
Dividing means subtracting: log (a ÷ b) = log a − log b. And a power comes to the front: log aⁿ = n log a.
Example: log10 25 + log10 4 = log10 (25 × 4) = log10 100 = 2. Be careful: log 25 + log 4 is NOT log 29.
🔑 Key idea: log ab = log a + log b, log (a ÷ b) = log a − log b, log aⁿ = n log a.
Evaluate log3 19.
Step 1: Write it as a power of 3
19 = 1 ÷ 32 = 3−2
Step 2: Read off the power
log3 19 = −2
Answer: −2
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